Question: Simplify the following expression and state the conditions under which the simplification is valid. You can assume that $k \neq 0$. $a = \dfrac{k - 1}{k - 7} \times \dfrac{k^2 + 3k - 70}{k^2 + 10k} $
Answer: First factor the quadratic. $a = \dfrac{k - 1}{k - 7} \times \dfrac{(k - 7)(k + 10)}{k^2 + 10k} $ Then factor out any other terms. $a = \dfrac{k - 1}{k - 7} \times \dfrac{(k - 7)(k + 10)}{k(k + 10)} $ Then multiply the two numerators and multiply the two denominators. $a = \dfrac{ (k - 1) \times (k - 7)(k + 10) } { (k - 7) \times k(k + 10) } $ $a = \dfrac{ (k - 1)(k - 7)(k + 10)}{ k(k - 7)(k + 10)} $ Notice that $(k + 10)$ and $(k - 7)$ appear in both the numerator and denominator so we can cancel them. $a = \dfrac{ (k - 1)\cancel{(k - 7)}(k + 10)}{ k\cancel{(k - 7)}(k + 10)} $ We are dividing by $k - 7$ , so $k - 7 \neq 0$ Therefore, $k \neq 7$ $a = \dfrac{ (k - 1)\cancel{(k - 7)}\cancel{(k + 10)}}{ k\cancel{(k - 7)}\cancel{(k + 10)}} $ We are dividing by $k + 10$ , so $k + 10 \neq 0$ Therefore, $k \neq -10$ $a = \dfrac{k - 1}{k} ; \space k \neq 7 ; \space k \neq -10 $